1. Field of the Invention
The present invention relates to a phase unwrapping method for a wrapped phase distribution obtained on a predetermined coordinate system in a fringe image analysis method for analyzing and measuring wavefront information concerning physical properties of a sample such as its form, thickness deviations, and refractive index distribution by using interference fringes, moiré fringes, or the like. In particular, the present invention relates to a technique for determining an unwrapping path on the coordinate system.
2. Description of the Prior Art
Various techniques have conventionally been known for measuring the surface form of a sample and the like with a high accuracy by analyzing fringe image signals of interference fringes, moiré fringes, and the like which are phase-modulated spatially or temporally. How to obtain accurate phase information from fringe image signals is a basic problem common in this kind of analysis techniques.
For example, in the heterodyne interference using Fourier transform, phase values obtained on a coordinate system corresponding to pixels on an imaging device are folded into a principal value range of [−π, π] (phase wrapping), whereby phase values become discontinuous for phases having a large dynamic range and exhibit indefinite values of integral multiples of 2π. Therefore, in order to attain a phase distribution in conformity to an actual surface form, phase unwrapping is necessary for determining the original continuous phase distribution from the phase distribution Φ(x, y) thus folded in the principal value range of [−π, π].
When carrying out such phase unwrapping, a highly accurate continuous phase distribution Φ′(x, y) can be determined even with a simple algorithm for smooth phase distributions yielding less noise, since the results of processing are not influenced by unwrapping paths (routes along with the phase unwrapping is carried out). When signals include a region yielding a large amount of noise and a low modulation (amount depending on interference fringe amplitude), results of processing totally differ from each other depending on unwrapping paths, which makes it necessary to choose an unwrapping path rationally on the coordinate system in order to determine the highly accurate continuous phase distribution Φ′(x, y).
Under such circumstances, techniques adapted to choose an unwrapping path rationally and carry out phase unwrapping favorably even when noise is high or modulation is low have recently been proposed.
Such a technique is a method in which portions yielding a higher contrast (higher modulation) in interference fringes are taken into account, unwrapping paths deemed to be the most rational are chosen successively from such portions, and phase unwrapping is carried out along thus chosen unwrapping paths. For choosing the most rational unwrapping paths, a graph problem known as minimum spanning tree problem in the field of computer science is adopted. Among such techniques, one known as amplitude maximum tree method has been known to be effective (see Proceedings of the 55th Annual Meeting of the Japan Society of Applied Physics (1994)).
In the amplitude maximum tree method, in the process of choosing unwrapping paths, it is necessary to successively carry out processing operations in which individual pixels on the unwrapping paths chosen heretofore and their adjacent pixels are multiplied by their respective recorded amplitudes so as to yield their products, thus determined individual products are stored in a memory of a computer and the like, those yielding the maximum value are chosen from these products, and a path between their corresponding pixels is taken as a new unwrapping path.
Recently, the number of pixels in imaging devices has been prone to increase dramatically, thereby enhancing demands for carrying out analysis with a higher accuracy by using fringe images having a greater number of pixels. To this aim, it is necessary that the processing speed of phase unwrapping be enhanced.
However, conventional phase unwrapping methods such as the amplitude maximum tree method use an algorithm in which the processing time increases as the number of pixels is greater so that the number of calculated values to be stored increases, whereby the processing time of phase unwrapping increases substantially in proportion to the increase in the number of pixels, which may be problematic in that it takes an enormous time for analyzing the phase state of fringe images.